zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Blow-up and global solutions for nonlinear reaction-diffusion equations with Neumann boundary conditions. (English) Zbl 1156.35407
Summary: The type of problem under consideration is $$\cases ((1+u) \ln^\alpha(1+u))_t= \nabla\cdot(\ln^\sigma(1+u)\nabla u)+(1+u) \ln^\beta(1+u),\quad &\text{in }D \times(0,T),\\ \frac{\partial u} {\partial n}=0,\quad &\text{on }\partial D \times(0,T),\\ u(x,0)=u_0(x) >0,\quad &\text{in }\overline D,\endcases$$ where $D\subset \bbfR^N$ is a bounded domain with smooth boundary $\partial D$, $N\ge 2$. It is proved that if $\beta-1>\sigma\ge\alpha\ge 0$, the positive solution $u(x,t)$ blows up globally in $\overline D$, whereas if $0\le\beta\le \sigma\le \alpha-1$, the positive solution $u(x,t)$ is global solution. Furthermore, an upper bound of the “blow-up time”, an upper estimate of the “blow-up rate”, and an upper estimate of the global solutions are given.

MSC:
35K60Nonlinear initial value problems for linear parabolic equations
35K57Reaction-diffusion equations
35K55Nonlinear parabolic equations
WorldCat.org
Full Text: DOI
References:
[1] Friedman, A.: Partial differential equation of parabolic type, (1964) · Zbl 0144.34903
[2] Galaktionov, V. A.; Kurdyumov, S. P.; Samarskii, A. A.: On approximate self-similar solutions of a class of quasilinear heat equations with a source, Math. USSR sbornik 52, 155-180 (1985) · Zbl 0573.35049 · doi:10.1070/SM1985v052n01ABEH002883
[3] Galaktionov, V. A.; Vazquez, J. L.: Regional blow-up in a semilinear heat equation with convergence to a Hamilton--Jacobi equation, SIAM J. Math. anal. 24, 1254-1276 (1993) · Zbl 0813.35033 · doi:10.1137/0524071
[4] Galaktionov, V. A.; Vazquez, J. L.: A dynamical systems approach for the asymptotic analysis of nonlinear heat equations, Proc. equa. Diff. 95 (1996) · Zbl 0884.35014
[5] Galaktionov, V. A.; Vazquez, J. L.: Blow-up for a quasilinear heat equation described by means of nonlinear Hamilton--Jacobi equations, J. differential equations 127, 1-40 (1996) · Zbl 0884.35014 · doi:10.1006/jdeq.1996.0059
[6] Galaktionov, V. A.; Vazquez, J. L.: The problem of blow-up in nonlinear parabolic equations, Discrete contin. Dyn. syst. 8, 399-433 (2002) · Zbl 1010.35057 · doi:10.3934/dcds.2002.8.399
[7] Galaktionov, V. A.; Vazquez, J. L.: A stability technique for evolution partial differential equations: A dynamical systems approach, (2004)
[8] Lacey, A. A.: Global blow-up of a nonlinear heat equation, Proc. roy. Soc. Edinburgh 104A, 161-167 (1986) · Zbl 0627.35047 · doi:10.1017/S0308210500019120
[9] Protter, M. H.; Weinberger, H. F.: Maximum principles in differential equations, (1967) · Zbl 0153.13602
[10] Samarskii, A. A.; Galaktionov, V. A.; Kurdyumov, S. P.; Mikhailov, A. P.: Blow-up in problems for quasilinear parabolic equations, (1987) · Zbl 1020.35001